Lagrange Error Bound to Find Error when using Taylor Polynomials

The theorem all looks a bit complicated but really it just boils down to taking some derivatives and then trying to maximize that function over a given interval. The rest of it is 'plug and chug'!

[sfiakhn]

Dear PatrickJMT,
You have saved our asses in every math class for four years and counting. Thank u for ur beautiful brain.
Sincerely
Sofia and sasha

[brodydrake]

when maximizing f(z) for sin are you maximizing it on any interval or only the given one (from 0 to 0.1)? Why did you plug in sin(pi/2) to get 1 instead of plugging in sin(0.1)?

[timhourigan]

Thank you for a concise, clear explanation. Most calculus textbooks do not offer near enough explanation or examples. Take a bow, sir! Bravo!

[gobletoffire]

Who else is studying for AP Calculus BC and IB Math HL?

[ddinggg]

Could you do another example, perhaps involving e?

[bjornnorenjobb]

This is the first time I felt that I started to get this, thank you so much! It would be so much appreciated if you did more videos on Lagrange Error Bound!!

[AishikGhosh]

thanks for continuing to make excellent videos for so many years. words can't describe how much you have helped me.

[miguelaphan]

this is the real classic video on this subject, so mahy videos came after this one, and as years pass by, still the classic

[mathematoy]

I dropped Degree in Biology for Math...thanks to my math doctor @patrickJMT. My math is surely salvated. Thanks.

[OSMAN-qdms]

Why is the max in sin (z) = 1 ? since z is max at 0.1, doesn't that mean that max in sin is sin(0.1)?

[calvinhyleung]

@patricJMT dude, thanks for making these video that prep me for the ap test!! you are my hero!!

[SequinBrain]

Minor point and I hate bringing these up, but, if (2n+1)=3, then n=1. Then also we're considering R_1, & f^(1+1), or f''(x). The process otherwise is similar. Other than that, for me, the hardest part of this is maximizing the z, or Xi in some texts. The rest is more or less plug-n-play.

[monsecortez]

I just have trouble grasping how to find the max value of z :(

[neylsonrodrigues]

Thank you. This was great and helps me so much on my homework

[hponde]

Patrick, considering that you estimated sin(0.1) to be at most 1 in order to get a bound on the error of sin(0.1), does that mean you can do the process again and get a better bound? And if so, how much can we improve this by just iterating this process? I'm sure it will converge, but to what value? That is a cute thing from the fact you use a 3rd degree approximation and the 4th derivative is the same function. Very interesting. Seems like a similar argument would hold for the exponential function too

[FnaticMedamri]

Does anyone notice that Patrick's voice has changed? I watched his calculus videos(which he uploaded 8 years ago) and I'm watching this video now and his voice changes are so obvious. A life spent in help of humanity. Thanks Patrick! You are saving our lives :P

[IAmTunnelRat]

Hey Patrick! I've been watching your videos for about 6 years now! I'm heading off to grad school and was wondering if you could do some GRE math videos. Thanks!

[cameronspalding]

The 4th degree polynomial is the same as the 3rd: we can find a smaller error bound: 0.1^5/5!

[azaz]

how do you see the stuff through ur hand when you are writing it..

[wellbangok]

Patrick do you have any videos on how to solve 3rd order Cauchy Euler equations? Thanks.